How to merge two objects at identical face?
I am not aware of a specific function that would align and merge at the same time so doing it manually is the best option that I know of.

First, to align the two objects select them both and in the properties panel (N) you will see rotation values, right click on them and click "Copy All To Selected" you should get something like this:



Then, under the viewport you should see a magnet button, enable the magnet and then set the snapping to vertex:



Select your "Plane" object and then press G after moving it in the general direction of the other object (its best to view it from the angle of the other face) the object will align to the vertex on the other shape:



After that switch into edit mode and keep the whole "Plane" object selected, then align one corner to the other and it will be perfectly aligned:



Finally, exit edit mode, select both objects and hit Ctrl+J, now go back into edit mode and you can edit both objects, simply merge the vertices together (This will be difficult but not impossible just click around the vertices and it will select both sides) and delete the inner face to clean up and you're done! :) Final result:

 

I would just like to say that although this method works it is completely unnecessary as these objects should've been created together in the first place, from what I saw there was no need for these to be separate objects, in future create objects in edit mode or extrude instead of creating new separate objects it will save you from going to all this effort and will be much faster too. Happy Blending :)